Question: Solve for $x$ and $y$ using elimination. $\begin{align*}9x-5y &= -8 \\ 4x-5y &= 2\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-9x+5y &= 8\\ 4x-5y &= 2\end{align*}$ Add the top and bottom equations. $-5x = 10$ Divide both sides by $-5$ and reduce as necessary. $x = -2$ Substitute $-2$ for $x$ in the top equation. $9( -2)-5y = -8$ $-18-5y = -8$ $-5y = 10$ $y = -2$ The solution is $\enspace x = -2, \enspace y = -2$.